Area of an equilateral triangle inscribed in a circle of radius a is

Options

A. `sqrt3 a ^(2) //4`
B. `pi a ^(2) //3`
C. `3 sqrt3 a ^(2) //4`
D. `sqrt3 pi a ^(2) //4`

Answer

To find the area of an equilateral triangle inscribed in a circle of radius \( a \), we can follow these steps:

### Step 1: Understand the relationship between the circle and the triangle
An equilateral triangle inscribed in a circle means that all three vertices of the triangle touch the circumference of the circle. The radius of the circle is the distance from the center of the circle to any vertex of the triangle.

### Step 2: Determine the side length of the triangle
For an equilateral triangle inscribed in a circle of radius \( a \), the relationship between the side length \( s \) of the triangle and the radius \( a \) is given by the formula:
\[ s = a \sqrt{3} \]
This is derived from the fact that the circumradius \( R \) of an equilateral triangle is related to its side length \( s \) by the formula:
\[ R = \frac{s}{\sqrt{3}} \]
Rearranging gives us \( s = R \sqrt{3} \).

### Step 3: Calculate the area of the triangle
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Substituting \( s = a \sqrt{3} \) into the area formula:
\[ A = \frac{\sqrt{3}}{4} (a \sqrt{3})^2 \]

### Step 4: Simplify the area expression
Calculating \( (a \sqrt{3})^2 \):
\[ (a \sqrt{3})^2 = a^2 \cdot 3 = 3a^2 \]
Now substituting this back into the area formula:
\[ A = \frac{\sqrt{3}}{4} \cdot 3a^2 \]
\[ A = \frac{3\sqrt{3}}{4} a^2 \]

### Final Result
Thus, the area of the equilateral triangle inscribed in a circle of radius \( a \) is:
\[ A = \frac{3\sqrt{3}}{4} a^2 \]

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